Biomedical Engineering Reference
In-Depth Information
μ
the magnetic permeability. The Maxwell's equations above have source terms given
by the charge density
where
E
is the electric field,
B
the magnetic field,
the dielectric permittivity, and
and the current density
J
. Additional contributions arise from
the time derivatives of the fields.
According to Eq. (
A.1
), the electric current density
J
is expressed as
ˁ
J
=
J
S
+
J
E
=
J
S
+
˃
E
,
(A.6)
where
J
E
is the “ohmic” current which flows in response to the local electric field, and
J
S
caused by the ionic flux
j
S
is the “source” current (or often called the “impressed”
current), which flows in response to transmembrane concentration gradients whose
sum corresponds to the net currents summed over all ions. In Eq. (
A.6
),
is the bulk
conductivity of the material, which is the main parameter governing the spread of
ionic currents through the volume. Variations in conductivity affect both EEG and
MEG spatial structure, although MEG is less sensitive to this parameter than EEG.
Charge conservation law is derived from Maxwell's equations in the following
manner. Taking the divergence of Eq. (
A.5
) and using Eq. (
A.2
), we obtain
˃
+
∂
∇·
E
+
∂ˁ
∂
∇·
J
=∇·
J
t
=
0
,
(A.7)
∂
t
∇·
(
∇×
)
=
where the relationship
0 is used. Integrating over a closed volume
V
bounded by a surface
S
and using the Gauss theorem yield:
B
=−
∂
∂
J
· ˆ
n
dS
V
ˁ
dV
,
(A.8)
t
S
where
n
is the outward unit normal to
S
. The integral on the left is the total current
flowing outward across the
S
, and the integral on the right is the total charge in
volume, so this relation states that the current flowing outward across
S
is equal to
the rate of change of the charge in the volume.
ˆ
A.2.1.1
Potential Formulation
It is convenient to re-express Maxwell's equations in terms of potential functions,
which are related to the fields by simple derivatives. Because
∇·
B
=
0, it is possible
to write
B
as a gradient of some other vector field
A
, such that
B
=∇×
A
,
(A.9)
where
A
is called the magnetic vector potential. Substituting the equation above into
Eq. (
A.4
), we obtain
